The simplest value of $$\frac{{3\sqrt 8 - 2\sqrt {12} + \sqrt {20} }}{{3\sqrt {18} - 2\sqrt {27} + \sqrt {45} }}{\text{is:}}$$

Solution (By Examveda Team)

$$\eqalign{ & {\text{Expression}} \cr & = \frac{{3\sqrt 8 - 2\sqrt {12} + \sqrt {20} }}{{3\sqrt {18} - 2\sqrt {27} + \sqrt {45} }} \cr & = \frac{{3\sqrt {2 \times 2 \times 2} - 2\sqrt {2 \times 2 \times 3} + \sqrt {2 \times 2 \times 5} }}{{3\sqrt {3 \times 3 \times 2} - 2\sqrt {3 \times 3 \times 3} + \sqrt {3 \times 3 \times 5} }} \cr & = \frac{{6\sqrt 2 - 4\sqrt 3 + 2\sqrt 5 }}{{9\sqrt 2 - 6\sqrt 3 + 3\sqrt 5 }} \cr & = \frac{{2\left( {3\sqrt 2 - 2\sqrt 3 + \sqrt 5 } \right)}}{{3\left( {3\sqrt 2 - 2\sqrt 3 + \sqrt 5 } \right)}} \cr & = \frac{2}{3} \cr} $$